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Bubble instabilities of mIIA on AdS$_4\times S^7$

by Pieter Bomans, Davide Cassani, Giuseppe Dibitetto and Nicolò Petri

This is not the latest submitted version.

This Submission thread is now published as SciPost Phys. 12, 099 (2022)

Submission summary

As Contributors: Pieter Bomans · Davide Cassani
Preprint link: scipost_202111_00014v1
Date submitted: 2021-11-08 16:31
Submitted by: Bomans, Pieter
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


We consider compactifications of massive IIA supergravity on a six-sphere. This setup is known to give rise to non-supersymmetric AdS$_4$ vacua preserving SO$(7)$ as well as G$_2$ residual symmetry. Both solutions have a round $S^6$ metric and are supported by the Romans’ mass and internal $F_6$ flux. While the SO$(7)$ invariant vacuum is known to be perturbatively unstable, the G$_2$ invariant one has been found to have a fully stable Kaluza-Klein spectrum. Moreover, it has been shown to be protected against brane-jet instabilities. Motivated by these results, we study possible bubbling solutions connected to the G$_2$ vacuum, representing non-perturbative instabilities of the latter. We indeed find an instability channel represented by the nucleation of a bubble of nothing dressed up with a homogeneous D2 brane charge distribution in the internal space. Our solution generalizes to the case where $S^6$ is replaced by any six-dimensional nearly-Kähler manifold.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 3 on 2022-2-8 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202111_00014v1, delivered 2022-02-08, doi: 10.21468/SciPost.Report.4359


The authors of this paper study the stability of a non-supersymmetric AdS$_4$ vacuum obtained from compactification of massive type IIA supergravity on $S^6$. There exists a $G_2$ invariant solution that is perturbatively stable and the authors set out to search for non-perturbative decay channels. They identify a bubbling solution that corresponds to such a non-perturbative decay channel via a bubble of nothing.

A previous report identified a few minor points for improvement and the authors addressed these points to the satisfaction of the other referee and myself.

The paper is now very well written and the result are very interesting, in particular, given the recent conjecture that all non-supersymmetric AdS vacua are unstable. The results also generalize to a larger class of solutions for which the internal $S^6$ is replaced by any other nearly Kähler manifold, which will be useful in the future. Therefore, in conclusion I recommend this paper for publication in SciPost.

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Anonymous Report 2 on 2022-1-25 (Invited Report)


After the clarifications and modifications on the original manuscript, I recommend the paper for publication in its current form.

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Anonymous Report 1 on 2022-1-12 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202111_00014v1, delivered 2022-01-12, doi: 10.21468/SciPost.Report.4172


A non-perturbative decay channel is proposed for the non-susy and G$_{2}$-invariant AdS$_{4}$ vacuum of massive IIA supergravity on S$^6$. This is further investigated using an effective supergravity description that retains, amongst others, two massive modes with normalised mass $m^2 L^2=20$ not included in the ISO(7) gauged supergravity. The non-perturbative solution describing the decay is obtained numerically and its geometry is qualitatively characterised in a piece-wise manner: and AdS region, a BoN region and a (smeared) D2-brane region. A non-flat but dS$_{3}$ slicing of the external spacetime geometry plays an essential role in each of the three regions, thus selecting very specific boundary conditions for the full solution to exist.

The manuscript contains interesting and timely results regarding the stability of non-susy AdS vacua, which is turning into an active research area in light of the Swampland conjecture. Before the paper can be recommended for publication in SciPost, I would like the authors to address the following comments/questions:

$\bullet$ It is stated at various places in the text that, for the analysed vacuum, a bubble is created and starts to expand, eventually consuming the complete vacuum. How does the existence of an intermediate (and only approximate) BoN-like region in the full solution guarantee this phenomenon? In other words, why having such an intermediate BoN-like region is enough to guarantee that the bubble has enough (physical) time to eat up the analysed vacuum?

$\bullet$ In the ansatz $(3.1)$, two functions $f_{41}(r)$ and $f_{42}(r)$ are introduced to specify $F_{4}$. Later on, when computing the dual $F_{6}=\star F_{4}$ in $(3.4)$, there is an additional function $f_{43}(r)$ appearing. And finally, when computing the Bianchi identity $dF_{4}=H_{3} \wedge F_{2}$ in (3.5), there is yet another function $\zeta(r)$ appearing. Are $f_{43}(r)$ and $\zeta(r)$ introduced somewhere in the text apart from the equations where they suddenly appear?

$\bullet$ Figure 4: I understand that the tuning of parameters in (4.13) is well motivated by the existence of a region with an almost constant $e^{3U-\phi/4}$ function. However, why is this sufficient to identify the resulting region with a BoN-regime? If looking closely at Figure 4, the claimed to be a BoN-like region between $r \sim 1.1$ and $r \sim 1.5$ is somehow not obvious. For example, the blue curve of the full solution and the green curve of the BoN are convex and concave within this window, respectively. Why the physical interpretation of this region as a BoN regime is justified and why the deviation from a truly BoN does not spoil the dynamical eating up process?

$\bullet$ Figure 4: The region of the full solution (blue curve) between $r \sim 0.9$ and $r \sim 1.1$ does not seem to be well approximated by any of the three regimes described above. What does it correspond to?

$\bullet$ There are $4$ coefficients $c_{1,2,3,4}$ compatible with regularity of the boundary conditions about the AdS region, and $2$ constraints in $(4.13)$ required by the intermediate bubble regime. So, naively, one would expect $4-2=2$ free parameters, although one of them could still be set at will by virtue of a further scaling of the radial coordinate. This would yet leave $1$ free parameter to perform a scanning of numerical solutions. Is the solution in $(4.14)$ just an element within a one-parameter family of solutions? If so, why this choice of solution and how do the other solutions look like? Or is this free parameter fixed by other means so the solution is unique?

$\bullet$ When discussing the possible existence of a similar flow for the susy and G$_2$-invariant vacuum the authors write: ``trying to impose similar conditions as those in $(4.13)$ will completely trivialize the
flow." What do they mean precisely by ``trivialize the flow"?

$\bullet$ Right afterwards the authors write: ``without such constraints one obtains solutions displaying an
intermediate regime that is still well approximated by $(2.5)$, but with $\alpha \neq \frac{1}{2}$". What value of $\alpha$ is actually obtained numerically and what is the physical interpretation of the resulting flow? Why, in general, an intermediate regime given by $(2.5)$ with $\alpha \neq \frac{1}{2}$ (thus not coming from Schw$_{5}$) does not represent a potential instability for a given asymptotic AdS solution (irrespective of susy)?

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Author:  Pieter Bomans  on 2022-01-24  [id 2119]

(in reply to Report 1 on 2022-01-12)

We thank the referee for the careful comments. We agree that some elements of our discussion are not entirely clear and can use some clarification. Below we address the different points raised by the referee (in the same order as they appear in the report), specifying the changes that we made in the manuscript.

1) Even though it is true that we do not have a concrete prescription for evaluating the on-shell Euclidean action (due to the presence of the source) and hence the decay rate of the putative vacuum, one could imagine reasoning along the lines of, where such a rate is estimated by evaluating the Euclidean action of the bubble background coupled to spherical brane probes. Moreover, we explicitly checked that a spherical bubble wall placed in AdS in our coordinate system, takes a finite amount of (global) time to reach the boundary, as judged by an observer at infinity describing AdS in global coordinates. We clarified this point in the discussion section by adding the sentence: "One can also check that the patch identified by the de Sitter slicing of AdS$_4$, which contains our bubble solution, reaches the boundary in finite global time".

2) Equation (3.4) indeed contains a typo, in this equation $f_{43}$ should be replaced by $f_{42}$. In equation (3.5), $f_6$, $f_{41}$ and $f_{42}$ are expressed in terms of the two new functions b and $\zeta$ in order to satisfy the Bianchi identity.

3) The second derivative of the full solution in the intermediate bubble regime indeed is different. However, as in this regime the slope of the bubble solution is almost constant, this is just a small perturbation caused by the influence of the other regimes of the solution. This therefore does not change the qualitative behavior of our solution.

4) The intermediate region between $r \sim 0.9$ and $r \sim 1.1$ corresponds to a gluing region describing the transition between the brane and bubble regimes. Since these are quite different, the gluing region is not approximated by either one of them.

5) As the referee correctly points out, in the input for our numerical integration there is one remaining free parameter, after reabsorbing the second one in the radial coordinate. However, for generic values of this parameter (with the only condition being that $\delta\phi >0$) the resulting numerical solution remains qualitatively the same and only differs in the explicit numerical values of the functions. As we observe under (4.13), the two free parameters should control the position of the brane source and the effective radius of the bubble. We clarified this point by adding the following footnote to the discussion below equation (4.13): "Given that the initial perturbation satisfies $\delta\phi>0$, the numerical integration for any value of these remaining free parameters will result in a qualitatively identical result."

6) We have clarified what we mean by "trying to impose similar conditions as those in (4.13) will completely trivialize the flow" by rephrasing it as: "trying to impose gluing conditions such that $(3U-\phi/4)$ is constant will completely fix the coefficients as $c_1=c_2=c_3=c_4=0$ instead of giving (4.13), which just corresponds to empty AdS$_4$.".

7) The actual value of \alpha obtained for the SUSY case is an irrational number different than 1/2. If one follows the idea of to evaluate the Euclidean action, then already the dilaton bubble with a value of \alpha other than 1/2 yields an infinite result.

Best regards, Pieter, Davide, Giuseppe and Nicolò

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